Hey, everyone. So not only can we add and subtract complex numbers, but we can also multiply them. Now, multiplication might seem a bit more intimidating than adding or subtracting, but we're actually still going to multiply our complex numbers the same way we multiply algebraic expressions. So, we're again going to take a skill that we already know and apply it to our complex numbers. And don't worry, I'm still going to walk you through it step by step. So let's go ahead and take a look. Now, algebraic expressions are always multiplied one of two ways, so complex numbers will be the same. We either distribute or we use foil. Now when we either distribute or foil, we're always going to end up with an I2 term. Now we know that I2 is just equal to negative one, so I'm going to use that to simplify my final answer. So let's go ahead and just jump right into an example. Looking at my first expression over here, I have 3I times 7 minus 2I. Now my very first step is going to be to either distribute or foil. And looking at my expression here, I have this 3I by itself. So since it's just one term, I'm multiplying this other complex number, I'm gonna go ahead and distribute. That seems like the best choice here. So, I'm gonna distribute this 3I into my 7 and my negative 2I in order to get 21I. And then 3I times negative 2I is gonna give me negative 6 I2. Make sure when you're multiplying an I by an I, you get I2. So step number 1 is done, and I can go ahead and move on to step 2, which is to apply the fact that I2 equals negative one. So looking at my expression here, I have 21I minus 6 I2. So I have my I2 right here, and I need to take this whole term and simplify it using I2 equals negative one. So this negative 6 I2 just becomes negative 6 times negative 1, and negative 6 times negative 1 is just positive 6. So I can bring my 21I down here, and I have 21I+6. So I've completed step 2. And step 3 is to combine my like terms. Now looking over here, I don't have any like terms that need to get combined. So step 3 is also done and this is my final answer. But I wanna make sure that I express my answer in standard form. So here at 21I+6, and I know that standard form is a+bi. So I'm gonna go ahead and flip this around in order to get 6 +21I, and this is gonna be my final solution. So let's go ahead and look at another example. So over here, I have negative 6 +2I times 3+4I. Let's go ahead and start with step 1, which is either to distribute or foil. Now since I have two complex numbers that each have two terms in them, it looks like foiling is gonna be my best option for step 1. So let's go ahead and foil. So I'm gonna start with my first term, so negative 6 times 3 is gonna give me negative 18. And then I have my outside terms, negative 6 times 4, which is gonna give me or times 4I, which is gonna give me negative 24I. Then I have my inside terms, 2I times 3 is gonna give me positive 6I. And then lastly, my last terms, which is 2I times 4I. Now this, since I'm multiplying two I terms, I'm gonna end up with an I squared. This is gonna give me plus 8 I2. Okay. So we have completed step number 1. Let's go ahead and move on to step 2, which is to apply that I2 equals negative one. Now I definitely have an I squared term over here. I have this 8I squared. So I'm gonna go ahead and simplify this whole term into 8 times negative one. Now 8 times negative one is just negative 8. So I have applied my I2 equals negative 1, and I can go ahead and pull all of my other terms down as well. So I can bring my negative 18, and then I have both of my I terms, negative 24I and positive 6I. So now step 3 is to combine all of my like terms, and I have some like terms that need to get combined here. So negative 18 and negative 8 are going to combine to give me negative 26. And then my other like terms are these I terms, negative 24 and positive 6I, which are gonna combine to give me negative 18I. So my final answer, I've completed step 3. I've combined my like terms. I have negative 26 minus 18I. And I, of course, wanna check that this is in standard form a+bi and it is. So I'm good to go and this is my final answer. That's all for this video. I'll see you in the next one.
Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
1. Equations & Inequalities
Complex Numbers
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